Comparison Among Three Harmonic Analysis Techniques on the Sphere and the Ellipsoid (original) (raw)
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Conventional spherical harmonic analysis for regional modelling of the geomagnetic field
Geophysical Research Letters, 1992
bstA••c!. Spherical Harmonic Analysis (SHA) is norreally used to model the three-dimensional global geomagnetic field. To address the same problem in regional modelling, Haines (1985) proposed Spherical Cap Harmonic Analysis (SCHA). This regional technique involves the computation of more complex Legendre functions with real (generally non-integer) harmonic degree. Here a new more practical technique is described; it is called Adjusted Spherical Harmonic Analysis (ASHA) because it is based on the expansion of conventional spherical harmonics after the colatitude interval is adjusted to that of a hemisphere. This kind of analysis can also be applied to modelling general two-dimensional functions. Introduction Haines (1985) introduced the elegant technique of Spherical Cap Harmonic Analysis (SCHA) to solve the problem of geomagnetic field regional modelling by means of spherical harmonic (SH) functions. When the usual system (R,O,A, i.e. radial distance, colatitude and longitude, respectively) is rotated to one (r,8,%) centred at the spherical cap which bounds the region of interest, and the new boundary conditions are applied to the basis functions which must be solutions of Laplace's equation, Haines found that the geomagnetic potential (and the assodated magnetic components) can be expanded in SHs with integer order m but non-integer degree r•t,: V(r,O,%) = a E E(alr) '""+•' h=0 m=0 ß gr cos + (co 0) {a is a reference radius, usually the Earth's mean radius, i.e. {11171.2 kin; n} is placed after the second summation because it depends on the value of m). Actu•dly the central role in the new technique is played by the colatitude basis functions, i.e. the "fractional" Legendre functions. Here we will shed some further light on this technique, illustrate what it means in practice and show how it is possible to simplify it; all by means of an approximate expansion of conventional SH functions. Some analogies with Fourier Analysis The main idea that motivated our work was the application of some techniques similar to those normally used in Fourier analysis. For example, to decrease the boundary effects ("leakage" in the frequency domain) in fitting general functions, a common procedure is to apply a cosine tapering (e.g. Bloomfield, 1976
Arabian Journal of Geosciences, 2013
The new global gravity models represented by global spherical harmonics like EGM2008 require a high degree and order in their coefficients to resolve the gravity field in local areas; therefore, there are interests to represent the regional or local field by less parameters and to develop a parameter transformation from the global model to a local kind of spherical harmonic model. The authors use local spherical cap harmonics for the regional gravity potential representation related to a local pole and a local spherical coordinate system. This allows to model regional gravity potential with less parameters and less memory requirements in computation and storage. From different kinds of representations of spherical cap harmonics, we have selected the so-called adjusted spherical cap harmonics (ASCH). This is the most appropriate for the presented mathematical model of deriving its coefficients from global gravity models. In that way, the global gravity models can fully be exploited and mapped to regional ASCH, in particular with respect to the computation of regional geoid models with improved solution.
Efficient Evaluation of Ellipsoidal Harmonics for Potential Modeling
arXiv: Numerical Analysis, 2017
Ellipsoidal harmonics are a useful generalization of spherical harmonics but present additional numerical challenges. One such challenge is in computing ellipsoidal normalization constants which require approximating a singular integral. In this paper, we present results for approximating normalization constants using a well-known decomposition and applying tanh-sinh quadrature to the resulting integrals. Tanh-sinh has been shown to be an effective quadrature scheme for a certain subset of singular integrands. To support our numerical results, we prove that the decomposed integrands lie in the space of functions where tanh-sinh is optimal and compare our results to a variety of similar change-of-variable quadratures.
Regional Gravity Field Modeling with Adjusted Spherical Cap Harmonics in an Integrated Approach
Schriftenreihe Fachrichtung Geodäsie der Technischen Universität Darmstadt 39, 2014
The main objective of this thesis is to develop an integrated approach for the computation of Height Reference Surfaces (HRS) in the context of GNSS positioning. For this purpose, the method of Digital Finite Element Height Reference Surface software (DFHRS) is extended, allowing the use of physical observations in addition to geometrical observation types. Particular emphasis is put on (i) using Adjusted Spherical Cap Harmonics to locally model the potential, (ii) developing a parameterization of coefficients for a least squares estimation, and (iii) optimizing the combination of data needed to calculate the coefficients. In particular, the selection of the terrestrial gravity measurements, height fitting points with known ellipsoidal and normal heights, and the use of the available global gravity models as additional observations are investigated. One of the main motivations is the need to compute a high precise local potential model with the ability to derive all components related to the potential W. These observation components are gravity , quasigeoid height , the geoid height , deflections of the vertical in the east and north direction ( ), the fitting points and the apriori information in terms of coefficients of a local potential model derived from the developed methods of a mapping of a global one. This thesis provides a method for local and global gravity and geoid modelling. The Spherical Cap Harmonics (SCH) for modeling the Earth potential are introduced in detail, including their relationship to the normal Spherical Harmonics (SH). The different types of Spherical Cap Harmonics, such as Adjusted Spherical Cap Harmonics (ASCH), Translated-Origin Spherical Cap Harmonics (TOSCH) and the Revised Spherical Cap Harmonics (RSCH) are discussed. The ASCH method was chosen in further for modeling the local gravitational potential due to its simple principle, that the integer degree and order Legendre functions are preserved and lead to faster implementation algorithms. The ASCH are used in this thesis to transform the global gravity models like EGM2008 or EIGEN05c to local gravity models, guaranteeing a much smaller number of coefficients and making the calculations faster and easier. Tests are applied to validate the use of ASCH for local gravity and potential modelling, with ASCH coefficients calculated in test areas. These coefficients were used to calculate the values of potential or the gravity for new points and then compared with the real measured values and reference values from global models. The tests include the transformation of global gravity models like EGM2008 and EIGEN05c to ASCH models and the integrated solution of heterogeneous groups of data including terrestrial gravity data, height fitting points and the locally mapped global gravity models. The region of the federal state of Baden-Württemberg in Germany was used as a test area for this thesis to prove the concept. Nearly 15000 terrestrially measured gravity observations were used to implement an ASCH model in degree and order of 300 in order to achieve a resolution of 0.01 mGal that corresponds to the measurement accuracy.
Spectral harmonic analysis and synthesis of Earth’s crust gravity field
Computational Geosciences, 2012
We developed and applied a novel numerical scheme for a gravimetric forward modelling of the Earth's crustal density structures based entirely on methods for a spherical analysis and synthesis of the gravitational field. This numerical scheme utilises expressions for the gravitational potentials and their radial derivatives generated by the homogeneous or laterally varying mass density layers with a variable height/depth and thickness given in terms of spherical harmonics. We used these expressions to compute globally the complete crustcorrected Earth's gravity field and its contribution generated by the Earth's crust. The gravimetric forward modelling of large known mass density structures within the Earth's crust is realised by using global models of the Earth's gravity field (EGM2008), topography/bathymetry (DTM2006.0), continental icethickness (ICE-5G), and crustal density structures (CRUST2.0). The crust-corrected gravity field is R. Tenzer(B) · V. Gladkikh obtained after modelling and subtracting the gravitational contribution of the Earth's crust from the EGM2008 gravity data. These refined gravity data mainly comprise information on the Moho interface and mantle lithosphere. Numerical results also reveal that the gravitational contribution of the Earth's crust varies globally from 1,843 to 12,010 mGal. This gravitational signal is strongly correlated with the crustal thickness with its maxima in mountainous regions (Himalayas, Tibetan Plateau and Andes) with the presence of large isostatic compensation. The corresponding minima over the open oceans are due to the thin and heavier oceanic crust.
A refined method of recovering potential coefficients from surface gravity data
Studia Geophysica et Geodaetica, 1990
S u m mar y: A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric boundary-value problem with a free boundarv is reduced to the problem with a fixed known telluroid. The main idea of the derivation consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained. Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the solutions of two linear problems utih'zing the fact that the non-linear boundary condition may be split into two parts; the linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the error data propagation is, consequently, maximally suppressed.
Journal of Geodesy, 2012
Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth's surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely-spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5): 282-289, 1997) for degree-360 models, produce cm-to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly-efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultrahigh resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points. Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200 m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth's surface.
Ju n 20 06 Spherical Harmonic Amplitudes From Grid Data
2006
The problem of resolving spherical harmonic components from numerical data defined on a rectangular grid has many applications, particularly for the problem of gravitational radiation extraction. A novel method due to Misner improves on traditional techniques by avoiding the need to cover the sphere with a coordinate system appropriate to the grid geometry. This paper will discuss Misner’s method and suggest how it can be improved by exploiting local regression techniques. PACS numbers: 04.25.Dm, 02.30.Mv, 02.30.Px, 02.60.Ed Harmonic Ampitudes From Grid Data 2
Computation of Gravity Field Functionals with a Localized Level Ellipsoid
Journal of Information System and Technology Management, 2021
This work is licensed under CC BY 4.0 The description of the earth's gravity field is usually expressed in terms of spherical harmonic coefficients, derived from global geopotential models. These coefficients may be used to evaluate such quantities as geoid undulations, gravity anomalies, gravity disturbances, deflection of the vertical, etc. To accomplish this, a global reference normal ellipsoid, such as WGS84 and GRS80, is required to provide the computing reference surface. These global ellipsoids, however, may not always provide the best fit of the local geoid and may provide results that are aliased. In this study, a regional or localized geocentric level ellipsoid is used alongside the EGM2008 to compute gravity field functionals in the state of Johor. Residual gravity field quantities are then computed using GNSS-levelled and raw gravity data, and the results are compared with both the WGS84 and the GRS80 equipotential surfaces. It is demonstrated that regional level ellipsoids may be used to compute gravity field functionals with a better fit, provided the zero-degree spherical harmonic is considered. The resulting residual quantities are smaller when compared with those obtained with global ellipsoids. It is expected that when the removecompute-restore method is employed with such residuals, the numerical quadrature of the Stoke's integral may be evaluated on reduced gravity anomalies that are smoother compared to when global equipotential surfaces are used
The role of high degree spherical harmonic model in local gravity field prediction
Artificial Satellites, 1987
The role of high degree spherical harmonic model as a reference field for local prediction of gravity functionals is shown. Three gravity field models: GEM 10B, Rapp 1978 and Rapp 1981 are used to compute geoid heights and gravity anomalies. The results are shown in the form of maps and profiles, and the choice of a proper reference model is discussed.